Related papers: An exact renormalization formula for Gaussian expo…
Gaussian Quadrature is a well known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest. In this paper we apply these ideas to…
The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum…
In the present paper we show how to obtain the well-known formula for Gauss sums and the Gauss reciprocity low from the Poison summarizing formula by using some ideas of renormalization and ergodic theories. We also apply our method to…
This paper presents a novel systematic methodology to obtain new simple and tight approximations, lower bounds, and upper bounds for the Gaussian Q-function, and functions thereof, in the form of a weighted sum of exponential functions.…
An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\theta)=\sum_{j=1}^{N} \exp (\pi ixj^2+2\pi ij\theta),$$ where $x$, $\theta$ are real and $N$ is a positive integer, is obtained as $x\rightarrow 0$ and…
In this paper we derive a new recovery procedure for the reconstruction of extended exponential sums of the form $y(t) = \sum_{j=1}^{M} \left( \sum_{m=0}^{n_j} \, \gamma_{j,m} \, t^{m} \right) {\mathrm e}^{2\pi \lambda_j t}$, where the…
Reconstructing a band-limited function from its finite sample data is a fundamental task in signal analysis. A Gaussian regularized Shannon sampling series has been proved to be able to achieve exponential convergence for uniform sampling.…
We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2\pi i\alpha n^k}$, for $k=1,2$, where $\alpha$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative,…
In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov's mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting…
In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for…
This paper provides a technique for evaluating some nonlinear Gaussian sums in closed forms. The evaluation is obtained from the known values of simpler exponential sums.
We derive in this short report the exact exponential decreasing tail of distribution for naturel normed sums of independent centered random variables (r.v.), applying the theory of Grand Lebesgue Spaces (GLS). We consider also some…
We derive in this article the exact non-asymptotical exponential and power estimates for self-normalized sums of centered independent random variables (r.v.) under natural norming. We will use also the theory of the so-called Grand Lebesgue…
Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to…
We discuss the difference Schr\"odinger equation $\psi_{k+1}+\psi_{k-1}+\lambda \cot(\pi\omega k+\theta)\psi_k=E\psi_k$, $k\in\mathbb{Z}$, where $\lambda$, $\omega$, $\theta$ and $E$ are parameters. We obtain explicit renormalization…
A proper formulation in the perturbative renormalization group method is presented to deduce amplitude equations. The formulation makes it possible not only avoiding a serious difficulty in the previous reduction to amplitude equations by…
A compact expression for the DeWitt-Schwinger renormalization terms suitable for use in even-dimensional space-times is derived. This formula should be useful for calculations of $<\phi^2(x)>$ and $<T_{\mu\nu}(x)>$ in even dimensions.
Let $G, G_1,\dots,G_N$ be independent copies of a standard gaussian random vector in $\mathbb{R}^d$ and denote by $\Gamma = \sum_{i=1}^N \langle G_i,\cdot\rangle e_i$ the standard gaussian ensemble. We show that, for any set $A\subset…
In this article we generalize the classical Edgeworth expansion for the probability density function (PDF) of sums of a finite number of symmetric independent identically distributed random variables with a finite variance to sums of…
The functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximated by a derivative expansion; no other approximation is made. This results in a set of coupled non-linear…